Planetary Precession

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Planetary Precession

• Jeremy Thornton

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Presentation Contents

• History of orbital precession
• Python Program
• Results
• View Python program in action

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History
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Mercury

• In elliptical orbits, the perihelion, point of
  closest approach, is a fixed position.
• Observed precession is 5600 (arc-seconds) per
  century (One arc-second is equal to 1/3600
  degree).
• This precession was discovered in the early 19th
  century and was subjected to Newtons Laws.

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Newton

• Predicted stable elliptical orbits for only
  planets in an ideal two-body system.
• Calculated that for Mercury, the Sun contributes
  5025 and surrounding planets contribute 532 per
  century to Mercurys precession.
• There was still 43 per century that Newtons
  laws left unaccounted for in Mercurys orbit.

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Einstein

• Developed the general theory of relativity.
• Used it to calculate relativistic precession for
  the planets in the inner solar system.
• Calculated that for Mercury relativistic
  precession due to gravitation being mediated by
  the curvature of spacetime accounted for 42.92
  per century.
• Today (2007) the relativistic precession for
  Mercury is recorded at 43.1 per century.

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PythonProgram
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Creation

• My first goal for the Python program was to
  create a working model of the solar system
  including Earth, the Sun, and Jupiter.
• In order to give smaller numbers to Python to
  increase the speed of the program, I chose to use
  a scale. My base unit of mass is one Sun, my
  base unit of distance is one AU and my base unit
  of time is one year.

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Forces

• I then used the leapfrog algorithm in combination
  with the Newtonian Inverse Square Law to set the
  force laws for the system.

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Initial Conditions

• The initial velocity for each planet gave the
  object a circular orbit by setting the velocity
  equal to vG/x. Since a perihelion is needed to
  observe precession, I added a small constant to
  the Earths velocity to make it an ellipse.

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Calculating Precession

• I set up the code to store the position of
  Earths maximal velocity (the perihelion) for
  each orbit. After its second orbit, I measured
  the angle ? between the position of the Earth in
  the 1st orbit and the position of the Earth in
  the 2nd orbit as seen in the diagram on the next
  page.

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Sun
Earth orbit 1
?
Earth orbit 2
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Collecting Results

• To get good data, I used dt equal to 0.1 days and
  ran the program over 50 orbits, measuring an
  average rate of precession (?/tOrbit) for varying
  mass of Jupiter.
• The next slides will show my program in action
  and explain what you are seeing. For the purpose
  of viewing, dt will be increased.

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Here is the standard orbit of the Earth in my
program. The green ring is the Earths orbit and
the blue ring is
Jupiters orbit. By setting dt equal to one day,
I can run the program for a long time and still
obtain reliable results and a good picture.
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I will now zoom in on the Earths orbit and
monitor it as the Earth precesses. The red
circles represent the
Earths position every year. The program starts
the Earth at 300. This is after 100 years and
the Earth has precessed slightly but is
relatively unnoticeable.
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As time passes, Python cannot keep track of so
many lines stacked on top of each other, so the
image is
distorted. The outer rim of the green represents
the Earths orbit and you can see after 1,000
years that the Earth has precessed significatnly
from its starting position.
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After 5,000 years you can see that the Earth has
precessed almost a quarter of the way around its
orbit.
It has been calculated that the Earths seasons
will shift about once every 5,000 years as you
can see here based on the Earths position.
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Results
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Results

• Data points recorded for Jupiters true mass and
  10, 25 ,50 ,75, and 100 times larger.
• As Jupiters mass increases, Earths precession
  rate increases linearly.
• If Jupiters mass becomes too large, the Earths
  orbit will become unstable and will eventually
  collide with Jupiter.

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What If

• The Earth was in a solar system with a large
  planet with mass 100 times greater than Jupiter?
• The Earth would precess around the Sun at the
  rate of about 0.05 radians per year.
• Over your lifetime 80 years, the Earth would
  precess almost 2/3 the way around the Sun.

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Python Program in Action
End Slide Show